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Section Day 2
This is an outline of the topics we covered in the second day of class. The skeleton notes are in a handout, which can be printed out using the printer icon at the top right of its section of the page for filling in during class. Filled notes for each day will be posted after class to Canvas.
Handout Thursday 5/21
Objectives: Advanced Learning Outcomes
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
State the following mathematical result: Socks-Shoes Property
State and instantiate the definitions of: order of a group, order of an element, subgroup, cyclic subgroup of a group generated by an element, center of a group
Given a generating set for a group, draw a Cayley graph for the group with respect to that generating set.
State the following mathematical results: One-Step Subgroup Test, Two-Step Subgroup Test
Determine whether a given subset of a group is a subgroup using the Subgroup Test.
Determine the cyclic subgroup generated by an element of a group.
Algebraist of the Day.
Joseph-Louis Lagrange , 1736-1813
Italian-French mathematician, once called “the greatest mathematician in Europe”
Laid groundwork for Galois’s work and worked on solving polynomials via
resolvents .
Convinced the Academy of Sciences to adopt the metric system for measurements
Group Properties. In the readings for today, you saw that the definition of a group quickly leads to three further properties.
Proposition 12 . Basic Group Properties.
Let \(G\) be a group. Then:
The identity of
\(G\) is unique.
\(G\) has right and left cancellation:
\(ab=ac\) or
\(ba=ca\) implies
\(b=c\) for all
\(a,b,c\in G\text{.}\)
Each element
\(a\in G\) has an
unique inverse, so we can unambiguously write
\(a^{-1}\text{.}\)
Proposition 13 . Socks-Shoes Principle.
Let
\(G\) be a group and
\(a,b\in G\text{.}\) Then
\((ab)^{-1}=b^{-1}a^{-1}\text{.}\)
Proof.
Example 14 . More Groups.
Let’s look at a few more group examples
Definition 15 . Subgroup.
A subset
\(H\) of a group
\(G\) is a
subgroup of
\(G\) if
\(H\) is also a group under the operation of
\(G\text{.}\) We write
\(H\leq G\) and if
\(H\neq G\) we call
\(H\) a
proper subgroup and may write
\(H\lt G\text{.}\)
Example 16 . Subgroup Examples.
Subgroups of
\(K_4\text{:}\)
There are two
subgroup tests , which parallel the
subspace tests you learned in linear algebra (because vector spaces are additive groups!).
Theorem 17 . Two-Step Subgroup Test.
Let \(G\) be a group and \(H\) be a nonempty subset of \(G\text{.}\) Then \(H\leq G\) if it satisfies
Closure under Operation: For any
\(a,b\in H\text{,}\) \(ab\in H\text{.}\)
Closure under Inverses: For any
\(a \in H\text{,}\) \(a^{-1} \in H\text{.}\)
Proof.
Theorem 18 . One-Step Subgroup Test.
Let
\(G\) be a group and
\(H\) be a nonempty subset of
\(G\text{.}\) Then
\(H\leq G\) if
\(ab^{-1}\in H\) for any
\(a,b\in H\text{.}\)
Example 19 . Using the Subgroup Tests.
Show that the set
\(H=\{e,r,r^2,r^3\}\) is a subgroup of
\(D_4\text{.}\)
Let
\(G\) be an Abelian group. Show that
\(H=\{g \in G \mid g^2=e\}\) is a subgroup of
\(G\)
Definition 20 . Center of a Group.
The center of a group \(G\) is
\begin{equation*}
Z(G)=\{x\in g \mid xg=gx \text{ for all } g\in G\}\text{.}
\end{equation*}
Theorem 21 . Center is a Subgroup.
\(Z(G)\) is a subgroup of
\(G\text{.}\)
Proof.
